Basin boundaries in asymmetric vibrations of a circular plate

In order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation, we reexamine a primary resonance, studied by Yeo and Lee [Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate, Journal of Sound and Vibration 257 (2002) 653-665] in which at most three stable steady-state responses (one standing wave and two traveling waves) are observed to exist. Further examination, however, tells that there exist at most five stable steady-state responses: one standing wave and four traveling waves. Two of the traveling waves lose their stability by Hopf bifurcation and have a sequence of period-doubling bifurcations leading to chaos. When the system has five attractors: three equilibrium solutions (one standing wave and two traveling waves) and two chaotic attractors (two modulated traveling waves), the basin boundaries of the attractors on the principal plane are obtained. Also examined is how basin boundaries of the modulated motions (quasi-periodic and chaotic motions) evolve as a system parameter varies. The basin boundaries of the modulated motions turn out to have the fractal nature.     

Experimental and numerical studies of noise-induced coherent patterns in a subexcitable system

A subexcitable medium of Belousov-Zhabotinsky (BZ) reaction subjected to external Gaussian white noise is studied in experiments and numerical simulations. We observe that at an optimal level of noise the wave sources of excited traveling waves become synchronous, as though there exists a long distance spatial correlation. The synchronous behavior fades if the noise level becomes larger or smaller. Numerical simulations confirm the experimental findings, and point out that the best synchronous behavior takes place when the signal-to-noise ratio of waves becomes largest.     

Direct observation of a "devil's staircase" in wave-particle interaction

We report the experimental observation of a "devil's staircase" in a time-dependent system considered as a paradigm for the transition to large-scale chaos in the universality class of Hamiltonian systems. A test electron beam is used to observe its non-self-consistent interaction with externally excited wave(s) in a traveling wave tube (TWT). A trochoidal energy analyzer records the beam energy distribution at the output of the interaction line. An arbitrary waveform generator is used to launch a prescribed spectrum of waves along the slow wave structure (a 4 m long helix) of the TWT. The resonant velocity domain associated to a single wave is observed, as well as the transition to large-scale chaos when the resonant domains of two waves and their secondary resonances overlap. This transition exhibits a "devil's staircase" behavior for increasing excitation amplitude, due to the nonlinear forcing by the second wave on the pendulum-like motion of a charged particle in one electrostatic wave.     

Dynamics of domain walls in pattern formation with traveling-wave forcing

We study dynamics of domain walls in pattern forming systems that are externally forced by a moving space-periodic modulation close to 2:1 spatial resonance. The motion of the forcing induces nongradient dynamics, while the wave number mismatch breaks explicitly the chiral symmetry of the domain walls. The combination of both effects yields an imperfect nonequilibrium Ising-Bloch bifurcation, where all kinks (including the Ising-like one) drift. Kink velocities and interactions are studied within the generic amplitude equation. For nonzero mismatch, a transition to traveling bound kink-antikink pairs and chaotic wave trains occurs.     

A simple model of ultrasound propagation in a cavitating liquid. Part II: Primary Bjerknes force and bubble structures

In a companion paper, a reduced model for propagation of acoustic waves in a cloud of inertial cavitation bubbles was proposed. The wave attenuation was calculated directly from the energy dissipated by a single bubble, the latter being estimated directly from the fully nonlinear radial dynamics. The use of this model in a mono-dimensional configuration has shown that the attenuation near the vibrating emitter was much higher than predictions obtained from linear theory, and that this strong attenuation creates a large traveling wave contribution, even for closed domain where standing waves are normally expected. In this paper, we show that, owing to the appearance of traveling waves, the primary Bjerknes force near the emitter becomes very large and tends to expel the bubbles up to a stagnation point. Two-dimensional axi-symmetric computations of the acoustic field created by a large area immersed sonotrode are also performed, and the paths of the bubbles in the resulting Bjerknes force field are sketched. Cone bubble structures are recovered and compare reasonably well to reported experimental results. The underlying mechanisms yielding such structures is examined, and it is found that the conical structure is generic and results from the appearance a sound velocity gradient along the transducer area. Finally, a more complex system, similar to an ultrasonic bath, in which the sound field results from the flexural vibrations of a thin plate, is also simulated. The calculated bubble paths reveal the appearance of other commonly observed structures in such configurations, such as streamers and flare structures.     

Sharp diffraction peaks from chaotic structures

Recently various models for spatially chaotic structures have been proposed. We study the diffraction patterns produced by plane chaotic waves incident on one-dimensional chaotic point scatterers. The spacing between the scatterers and the dynamics of the incident wave are given by a logistic map or standard map. We find a sharp diffraction peak when the incident dynamics is produced by the same map as the structure of the spatial configuration. The diffraction pattern is symmetric about the incident direction only if the map dynamics is invertible. Diffraction patterns with chaotic incident waves have a large signal-to-noise ratio and are well suited for pattern identification. We discuss possible applications to the scattering of microwaves from aperiodic structures. (c) 1997 American Institute of Physics.     

Phonon-polariton waves on the surface of SiC crystal

The excitation and propagation of traveling phonon-polariton waves on the surface of silicon carbide (SiC) excited by light at a frequency close to the lattice resonance have been investigated. These waves are excited in the presence of the boundary of a metal mask deposited on the crystal surface. The use of the Green’s function has been shown to provide good qualitative agreement with the observed distribution of the amplitude and phase of the field on the surface under these experimental conditions. It has been shown that only consideration of the mask boundary as an extended source of traveling surface waves, which removes the inhibition of the generation of waves in the noncoincidence of the wave vector, cannot quantitatively describe the phenomenon. The spatial resolution of the used scanning near-field optical microscope is no worse than 150 nm at a wavelength of 10 μm.     

Elimination of Anti-spiral Waves by Local Inhomogeneity in Oscillatory Systems

采用系统中的局部不均匀性消除振荡系统中的反螺旋波. 该不均匀性在系统中成为一个波源,不断产生稳定的相波. 研究结果发现,不均匀性的尺寸大小存在一个临界值,如果低于此临界值,则系统将无法激发任何相波. 根据不均匀性形状的不同,系统分别产生靶波和行波. 此外,实验还发现靶波与反螺旋波之间以及行波与反螺旋波之间存在着不同的动态竞争. 数值计算表明,对于行波,无论是低频行波还是高频行波,都可以成功地消除系统中的反螺旋波;而对于靶波,只有低频靶波才可以消除反螺旋波. 此控制方法简单易行,且同样适用于消除向外传播的螺旋波     

Modulation instability analysis,optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.     

Chaotic particle motion under linear surface waves

We investigate the motion of infinitesimal particles in the flow field inside the fluid under a traveling surface wave. It is shown that, even for two-dimensional waves, a superposition of two or more traveling harmonic waves is enough to generate chaotic particle motion, i.e., Lagrangian chaos. (c) 1996 American Institute of Physics.     

Scale and space localization in the Kuramoto-Sivashinsky equation

We describe a wavelet-based approach to the investigation of spatiotemporally complex dynamics, and show through extensive numerical studies that the dynamics of the Kuramoto-Sivashinsky equation in the spatiotemporally chaotic regime may be understood in terms of localized dynamics in both space and scale (wave number). A projection onto a spline wavelet basis enables good separation of scales, each with characteristic dynamics. At the large scales, one observes essentially slow Gaussian dynamics; at the active scales, structured "events" reminiscent of traveling waves and heteroclinic cycles appear to dominate; while the strongly damped small scales display intermittent behavior. The separation of scales and their dynamics is invariant as the length of the system increases, providing additional support for the extensivity of the spatiotemporally complex dynamics claimed in earlier works. We show also that the dynamics are spatially localized, discuss various correlation lengths, and demonstrate the existence of a characteristic interaction length for instantaneous influences. Our results motivate and advance the search for localized, low-dimensional models that capture the full behavior of spatially extended chaotic partial differential equations. (c) 1999 American Institute of Physics.     

单向耦合神经元网络中螺旋波的自发形成机制

构造一个具有单向耦合的二维神经元网络,引入信息传输熵来描述定向信息传输,采用Hindmarsh-Rose神经元模型研究网络中螺旋波等有序波自发产生的机制.数值模拟表明：适当选取耦合的强度和单向耦合的距离,网络可自发出现螺旋波、行波、靶波和平面波.各种有序波的产生与网络中出现信息间歇定向传输有关,网络出现单或多螺旋波时发生熵共振现象.噪声、抑制性耦合和排斥性耦合诱发螺旋波时网络中也存在信息间歇定向传输.首次发现自维持长平面波,其存在是由于网络存在持续的强信息定向传输.     

Dynamics of traveling waves in the transverse section of a laser

We analyze the general features of the formation and interaction of transverse traveling waves and the appearance of filamentation in broad area semiconductor lasers with current profiling. For small apertures, the emitted profile is symmetric consisting of two counterpropagating transverse traveling waves, both emanating from the center of the device. For larger apertures, the emission becomes asymmetric as one of the traveling waves expands to occupy an increased area while the other occupies the remaining, smaller spatial region. In both devices, the pattern becomes unstable at higher injection currents due to optical filamentation, although an intermediate state is present in the wider device whereby the dominant wave undergoes a Hopf bifurcation before filamentation occurs.     

Excitation and sensing of multiple vibrating traveling waves in one-dimensional structures

Lightly damped vibrating structures normally exhibit vibration patterns that are a combination of standing waves, i.e. mode shapes. Traveling waves, on the other hand, occur only under special circumstances. In this work, the theoretical conditions under which traveling waves prevail in finite structure are investigated. These conditions are highly sensitive to the geometrical and material parameters of the structure and in particular the vibration pattern is sensitive to the boundary conditions. There are several combinations under which traveling waves cannot be formed and these ill-posed cases are analyzed in some detail. To overcome the unavoidable uncertainties in a model, a tuning process based on identification and optimization of the excitation is suggested. The identification process uses a parametric algorithm to estimate the wavenumbers of the measured vibrations. Then, the waves are decomposed into traveling and standing parts and the external excitation is tuned until a pure traveling wave is formed.     

二维三组元声子晶体中的半狄拉克点及奇异特性

设计了一种二维三组元声子晶体结构, 利用偶然简并的方法在布里渊区中心实现了半狄拉克点, 探究了随着组元几何参数的改变半狄拉克点的演变过程. 利用有限元方法研究发现在半狄拉克点频率附近沿着ΓX 方向声子晶体表现出与零折射率材料相似的行为, 许多奇异特性如单向透射等均可以观察到. 另外还发现在半狄拉克点频率附近声子晶体是各向异性的, 沿着不同方向声波的传播现象是不同的, 这种特性是狄拉克点及类狄拉克点所不具备的. 这种各向异性的声波传播特性有许多重要的应用, 如单方向完美透射和单方向波前整形等.     

A normal form for excitable media

We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.     

Resonant filtering of compositional waves in multicellular networks

Theoretically predicting the stationary and traveling compositional wave patterns observed over trivial multicellular topologies is common place in biology. Systematic methods are needed to find the relationships between the forms of such waves and the underlying network topologies. Here, we introduce one such method based on the Interface Response Theory (IRT) that combines models of intracellular biochemical reactions with those of intercellular networks to analytically track time variations of concentration profiles across nontrivial cellular network topologies. Cellular chains of infinite length are shown to sustain traveling planar waves under certain conditions among the coefficients of the intracellular chemical reactions. A non-trivial cellular network composed of a finite length side chain attached to the infinite backbone leads to the formation of a standing wave pattern upstream from the site of attachment. Downstream, the wave proceeds although a limited number of frequency bands are filtered out of the original content with that number being equal to the number of cells in the side chain.     

Relaxation of the director field in the form of a traveling wave in twisted nematic cells

The orientational relaxation of the director field toward its equilibrium orientation in the form of a traveling wave is investigated. It is shown that the nonlinear relaxation can occur in the form of a traveling wave in a twisted nematic cell under specific conditions for the external electric field and material parameters characterizing the nematic liquid crystal. The calculations of the relaxation processes in the form of traveling waves have demonstrated that these nonlinear processes in the twisted nematic cell proceed more easily when the initial perturbation of the director is induced in the vicinity of one of the bounding cell surfaces.